One of the Territorial Army soldiers who died on the Brecon Beacons on Saturday was a Maths teacher from Lewisham.
Lance Corporal Craig Roberts was taking part in a military training exercise, believed to be part of selection for the TA's section of the SAS.
More top news
In a wide-ranging Peston on Sunday interview, he said Parliament security needs reviewing and the PM's counter-terror strategy is flawed.
The terror attack carried out by Khalid Masood at Westminster that left four people dead lasted less than two minutes say police.
Two knives were found at the scene in Islington, North London, and the suspects fled on foot before being arrested.